Private Equity puzzles, solved step by step

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  1. 060You have nine gold bars that look identical, but one is slightly lighter than the rest. Using a balance scale, what is the fewest number of weighings that is guaranteed to find the light bar?Logic and brainteasersWarm upMid-market buyout fund

    Try it first

    How many weighings guarantee you find the light bar?

    Show the worked solution

    Two weighings. Put three bars on each pan and three aside. If one pan rises, the light bar is among those three; if they balance, it is among the three aside. Then take that group of three and weigh one bar against one: the pan that rises holds it, and a balance means it is the bar left out. Each weighing has three outcomes, so two weighings tell nine bars apart.

    Why split into three groups and not two?

    Think of a balance scale as a question with three possible answers: left is lighter, right is lighter, or they balance. Asking a yes or no question wastes one of those answers. A weighing that splits the suspects into three equal groups uses every outcome, so each weighing cuts the suspects to a third rather than a half. Nine bars become three after one weighing and one after two.

    Each weighing has three outcomes, so it can cut the suspects to a thirdWeighing 1: three against three, three asideleft panright panasideLeft riseslight bar in left 3Right riseslight bar in right 3Balancelight bar in aside 3Weighing 2: from those three, one against one, one asidepan Apan BasideA rises: A is lightB rises: B is lightBalance: the aside bar3 outcomes x 3 outcomes = 9 bars told apart in 2 weighings
    The first weighing puts three bars on each pan and three aside, and each of its three outcomes leaves three suspects; the second weighing, one against one with one aside, picks the light bar out of those three, so two weighings cover all nine bars.

    How do you prove that one weighing is not enough?

    Count the answers a single weighing can give: three. There are nine bars, so nine different possible answers to the question which bar is light. One weighing can distinguish at most three cases, two weighings at most nine, so with nine bars two is both enough and the minimum. The same count gives the general rule: n weighings can find one light bar among up to 3 to the power n bars, so three weighings handle 27.

    The relationship
    3w≥N⇒w=⌈log⁡3N⌉=⌈log⁡39⌉=23^{w} \geq N \quad\Rightarrow\quad w = \lceil \log_3 N \rceil = \lceil \log_3 9 \rceil = 2
    wthe number of weighings
    Nthe number of bars, 9
    3outcomes per weighing: left light, right light, balance
    What it says in wordsYou need enough weighings that three to the power of the weighings covers every bar.

    Where candidates lose it

    The common wrong route is halving: four against four with one aside. If the pans balance you are done in one, but if they do not you have four suspects, which take two more weighings, three in the worst case. The question asks for a guarantee, so the worst case is what counts.

    The second miss is getting two by luck and being unable to say why it is the minimum. Give the counting argument: one weighing has three outcomes and cannot separate nine bars.

    What the interviewer asks next

    • What if you have 12 bars and the odd one could be heavier or lighter?
    • With three weighings, what is the most bars you can handle?
    • Where in diligence do you split a problem into three rather than two?
  2. 080A bat and a ball cost Rs 110 in total. The bat costs Rs 100 more than the ball. How much does the ball cost?Logic and brainteasersWarm upMid-market buyout fundIndian mid-market PE

    Try it first

    Answer inside five seconds.

    Show the worked solution

    The ball costs Rs 5 and the bat Rs 105. Call the ball x. The bat is x plus 100, so together they are 2x plus 100, which must equal 110. That makes 2x equal to 10 and x equal to 5. The fast answer of Rs 10 fails the check: a Rs 100 bat is only Rs 90 more than a Rs 10 ball.

    Why does Rs 10 jump out, and why is it wrong?

    The numbers are built so that 110 minus 100 hands you 10 without thinking. That subtraction answers a different question: what is left if the bat costs exactly 100. The condition is a difference, the bat is 100 more than the ball, not a price, so the bat must contain a whole ball's worth plus 100. Check Rs 10 against the condition and it fails at once: 100 minus 10 is 90.

    The bat is a ball plus 100, so two balls plus 100 make 110Fast answer101090batball= 110Bat 100 is only 90 more than the ballWritten out55the extra 100batball= 110Bat 105 is exactly 100 more than ball 5x + (x + 100) = 110, so 2x = 10 and x = 5
    The fast answer of a Rs 10 ball and a Rs 100 bat leaves the bat only Rs 90 more than the ball; splitting the bat into a ball-sized piece plus Rs 100 shows two balls plus 100 make 110, so the ball is Rs 5 and the bat Rs 105.

    What is the interviewer actually testing?

    Not algebra. They are testing whether you check a fast answer against the conditions before you say it. A deal model full of quick numbers has the same risk: a cell that looks right because it is round, never tested against the constraint it was meant to meet. Writing one line, x plus x plus 100 equals 110, takes three seconds and removes the risk.

    Say the answer, then the check in the same breath: ball 5, bat 105, difference 100, total 110. Interviewers often use this as a warm-up and judge you more on the check than on the number.

    Where candidates lose it

    Rs 10 is the whole trap, and quick, confident candidates say it most often. The interviewer is not looking for speed here; they want to see a pause and a check.

    If you do say Rs 10, recover by checking out loud: then the bat is 100, which is only 90 more. Correcting yourself in the room scores far better than defending the wrong number.

    What the interviewer asks next

    • A bat and ball cost Rs 1,100 and the bat costs Rs 1,000 more. What is the ball?
    • Where in an LBO model does a fast round number most often hide an error?
    • If the bat costs three times the ball and the total is Rs 110, what is each?